Quantum Topology Seminar
Matthew Harper
The Ohio State University
A Generalization of the Alexander Polynomial from Higher Rank Quantum Groups
Abstract: Murakami and Ohtsuki have shown that the Alexander polynomial is an $R$-matrix invariant associated with representations $V(t)$ of unrolled restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. In this context, the highest weight $t\in\mathbb{C}^\times$ of the representation determines the polynomial variable. In this talk, we discuss an extension of their construction to a link invariant $\Delta_{\mathfrak{g}}$, which takes values in $n$-variable Laurent polynomials, where $n$ is the rank of $\mathfrak{g}$.
We begin with an overview of computing quantum invariants and of the $\mathfrak{sl}_2$ case. Our focus will then shift to $\g=\slthree$. After going through the construction, we briefly sketch the proof of the following theorem: For any knot $K$, evaluating $\Delta_{\mathfrak{sl}_3}$ at ${t_1=\pm1}$, ${t_2=\pm1}$, or ${t_2=\pm it_1^{-1}}$ recovers the Alexander polynomial of $K$. We also compare $\Delta_{\mathfrak{sl}_3}$ with other invariants by giving specific examples. In particular, this invariant can detect mutation and is non-trivial on Whitehead doubles.
Based on arxiv.org/abs/1911.00641 and arxiv.org/abs/2008.06983.
Thursday September 10, 2020 at 4:00 PM in Zoom